orthogonal

(adjective)

statistically independent, with reference to variates

Related Terms

  • two-way ANOVA
  • homoscedastic
  • least squares
  • polynomial regression

Examples of orthogonal in the following topics:

  • Experimental Design

    • Orthogonality: Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out.
    • Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal.
    • Because of this independence, each orthogonal treatment provides different information to the others.
    • If there are $T$ treatments and $T-1$ orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
    • Outline the methodology for designing experiments in terms of comparison, randomization, replication, blocking, orthogonality, and factorial experiments
  • Specific Comparisons (Independent Groups)

    • Independent comparisons are often called orthogonal comparisons.
    • There is a simple test to determine whether two comparisons are orthogonal: If the sum of the products of the coefficients is 0, then the comparisons are orthogonal.
    • Therefore, the two comparisons are orthogonal.
    • Table 7 shows two comparisons that are not orthogonal.
  • Specific Comparisons (Correlated Observations)

    • Although mathematically possible, orthogonal comparisons with correlated observations are very rare.
  • Two-Way ANOVA

    • Each level of one factor is tested in combination with each level of the other(s), so the design is orthogonal.
  • Tests Supplementing ANOVA

    • Important issues concerning multiple comparisons and orthogonal comparisons are discussed in the Specific Comparisons section in the Testing Means chapter.
    • Naturally, the same consideration regarding multiple comparisons and orthogonal comparisons that apply to other comparisons among means also apply to comparisons involving components of interactions.
  • Polynomial Regression

    • Although the correlation can be reduced by using orthogonal polynomials, it is generally more informative to consider the fitted regression function as a whole.
  • Exercises

    • True/false: The contrasts (-3, 1 1 1) and (0, 0 , -1, 1) are orthogonal.
  • Glossary

    • When comparisons among means provide completely independent information, the comparisons are called "orthogonal. " If an experiment with four groups were conducted, then a comparison of Groups 1 and 2 would be orthogonal to a comparison of Groups 3 and 4 since there is nothing in the comparison of Groups 1 and 2 that provides information about the comparison of Groups 3 and 4.
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